This is typically the mask worn for health purposes and you can either 하다 or 쓰다 it
Decorative masks. These tend to be hard and thick and you 쓰다 them
Decorative masks that are typically made out of fabric and you 쓰다 them. Unlike 가면, these masks are soft and thin/pliable. (I saw the ski mask ones referred to as 복면 마스크 and 복면 모자)
eyepatch/sleepmask and you can 하다, 끼다, 차다, or 쓰다 them
착용하다 is more of a formal verb for “wear” and is used for anything you wear on the body. so you can use it for all of these, but in normal everyday situations you wouldnt use it.